Numerical Methods For Option Pricing With Jumps

The pricing of financial derivatives is one of the most important issues in finan-cial research in recent decades,which promotes the development of global financial market.As one of the financial derivatives,option pricing becomes particularly im-portant.Since the emergence of Black-Scholes option pricing theory,the pricing of European option,American option,Asian option and barrier option pricing based on Black-Scholes model have been deeply studied and developed.With the deepening of research,the assumption of continuous change of underlying asset and constant volatility is no longer applicable to the changes in the actual financial market.In order to better describe the price fluctuations of the underlying asset,a series of alternative models,such as jump-diffusion model,switching model and stochastic volatility mod-el,have emerged.Therefore,in this paper,we focus on option pricing with jumps and study the numerical solutions of the jump-diffusion model,the stochastic volatility model with jumps and the regime-switching jump-diffusion model.The details are as follows:When the underlying asset follows jump-diffusion process,the price of a Eu-ropean option satisfies a partial integral differential equation.Since the equation contains non-local integral term,it brings some difficulties to the numerical calcula-tion of the model.The implicit-explicit method is used to the discretization time,which can not only reduce the computation cost,but also ensure the stability of the numerical scheme.On the basis of the existing research,we use the implicit-explicit two-step backward differentiation formula with variable step-sizes for time discretiza-tion,and prove the stability,consistency and convergence of the method.Due to the non-smoothness of the initial data,the convergence rate of the numerical solution near the strike price may be reduced.Therefore,we adopt local mesh refinement strategy in space.In addition,for the linear complementarity problem caused by American options,it is converted into a partial integral differential equation by in-troducing penalty parameter ?.Finally,the numerical results verify the correctness of the theoretical analysis.When the underlying asset follows jump-diffusion process,and the volatility of the underlying asset is random,the Bates model satisfied by the European option is a two-dimensional partial integral differential equation.We combine the high-order compact method in space with splitting method in time,and obtain the time splitting high-order compact scheme for the Bates model.This shows that the scheme can achieve fourth-order accuracy in space and second-order accuracy in time.And we analyze the stability of time splitting high-order compact scheme.Numerical results verify the effectiveness of the proposed method.When the underlying asset follows regime-switching jump-diffusion process,the mathematical model of European option price satisfies partial integral differential equations,and the computation amount is very large.We use implicit-explicit BDF2 method for time discretization,and prove that the time semi-discretization method is L2 stable,and has a second order accuracy.A fourth-order compact finite difference scheme is used for spatial discretization.Because of the discontinuity of the initial data,the convergence rate of the numerical solution can not reach fourth-order accu-racy in space.Therefore,a local mesh refinement strategy is adopted near the strike price,so thatthe accuracy can achieve fourth order.Finally,the numerical results verify the correctness of the theoretical results.Finally,we consider the numerical method for general stochastic differential e-quation with Markovian switching and jumps.When the coefficients of the equation satisfy local Lipschitz condition.nd linear growth condition or local Lipschitz condi-tion and one-sided Lipschitz condition,we prove the existence and uniqueness of the solution to the equation,and obtain the boundedness of the p(p?1)-order moment under one-sided Lipschitz condition and global Lipschitz condition.In addition,it is proved that when the drift coefficient satisfies one-sided Lipschitz condition and polynomial growth condition,the diffusion coefficient and jump coefficient satisfy the global Lipschitz,the strong convergence rate of the backward Euler scheme is arbi-trarily close to 1/2 order.Numerical examples further illustrate the correctness of the theoretical analysis.